= Renormalized interaction potential
{title2=$V_{\mathrm{ren}}(x)=\int[W(x-z)-W(-z)]\,d\mu(z)$}
For a twice continuously differentiable <interaction potential> with $\|D^2W\|\leq L$ and a <probability measure> with finite first <absolute moment>, this subtraction produces a finite potential even if $W*\mu$ is infinite. Taylor's formula and linear growth of the <gradient> give
$$
|W(x-z)-W(-z)|\leq\bigl(|\nabla W(0)|+L|z|\bigr)|x|+\frac L2|x|^2.
$$
Dominated <differentiation under the integral sign> gives $\nabla V_{\mathrm{ren}}=(\nabla W)*\mu$ and $D^2V_{\mathrm{ren}}=(D^2W)*\mu$. If the unrenormalized <convolution> is finite, the two potentials differ only by a constant, so they generate the same <velocity field> when that field depends on their <gradient>.
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